The main topic of this talk is a version of the Craig interpolation theorem formulated for an arbitrary logical system formalised as an institution, which proved crucial in the development of a number of key ideas and results concerning foundations of software specification and formal development. I will examine how admitting empty carriers in many-sorted first-order structures affects (or rather, does not affect) Craig interpolation and a number of other classical model-theoretic results. Then, more generally, I will discuss preservation of interpolation properties under institution extensions by new models and sentences, pointing out that some interpolation properties remain stable under such extensions, even if quite arbitrary new models and sentences are permitted. I will also present (time permitting) complete characterisations of such situations for institution extensions by new models, by new sentences, as well as by new models and sentences, respectively.